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Mirrors > Home > ILE Home > Th. List > r19.3rm | GIF version |
Description: Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 19-Dec-2018.) |
Ref | Expression |
---|---|
r19.3rm.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
r19.3rm | ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2200 | . . 3 ⊢ (𝑎 = 𝑦 → (𝑎 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
2 | 1 | cbvexv 1890 | . 2 ⊢ (∃𝑎 𝑎 ∈ 𝐴 ↔ ∃𝑦 𝑦 ∈ 𝐴) |
3 | eleq1 2200 | . . . 4 ⊢ (𝑎 = 𝑥 → (𝑎 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴)) | |
4 | 3 | cbvexv 1890 | . . 3 ⊢ (∃𝑎 𝑎 ∈ 𝐴 ↔ ∃𝑥 𝑥 ∈ 𝐴) |
5 | biimt 240 | . . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑))) | |
6 | df-ral 2419 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
7 | r19.3rm.1 | . . . . . 6 ⊢ Ⅎ𝑥𝜑 | |
8 | 7 | 19.23 1656 | . . . . 5 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑)) |
9 | 6, 8 | bitri 183 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 𝑥 ∈ 𝐴 → 𝜑)) |
10 | 5, 9 | syl6bbr 197 | . . 3 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) |
11 | 4, 10 | sylbi 120 | . 2 ⊢ (∃𝑎 𝑎 ∈ 𝐴 → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) |
12 | 2, 11 | sylbir 134 | 1 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1329 Ⅎwnf 1436 ∃wex 1468 ∈ wcel 1480 ∀wral 2414 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-cleq 2130 df-clel 2133 df-ral 2419 |
This theorem is referenced by: r19.28m 3447 r19.3rmv 3448 r19.27m 3453 indstr 9381 |
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